Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \sin y + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x * math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \sin y + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x * math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \sin y, z \cdot \cos y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x (sin y) (* z (cos y))))

C

double code(double x, double y, double z) {
	return fma(x, sin(y), (z * cos(y)));
}

Julia

function code(x, y, z)
	return fma(x, sin(y), Float64(z * cos(y)))
end

Wolfram

code[x_, y_, z_] := N[(x * N[Sin[y], $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]
2. Step-by-step derivation

   1. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)} \]

4. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(x, \sin y, z \cdot \cos y\right) \]

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \sin y + z \cdot \cos y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (sin y)) (* z (cos y))))

C

double code(double x, double y, double z) {
	return (x * sin(y)) + (z * cos(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * sin(y)) + (z * cos(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.sin(y)) + (z * Math.cos(y));
}

Python

def code(x, y, z):
	return (x * math.sin(y)) + (z * math.cos(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * sin(y)) + Float64(z * cos(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * sin(y)) + (z * cos(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]

2. Final simplification 99.8%

   \[\leadsto x \cdot \sin y + z \cdot \cos y \]

Alternative 3: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sin y\\ t_1 := z \cdot \cos y\\ \mathbf{if}\;y \leq -6.1 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;-0.5 \cdot \left(y \cdot \left(y \cdot z\right)\right) + \left(z + x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (sin y))) (t_1 (* z (cos y))))
   (if (<= y -6.1e+147)
     t_0
     (if (<= y -1.15e+103)
       t_1
       (if (<= y -4000000.0)
         t_0
         (if (<= y 0.035)
           (+ (* -0.5 (* y (* y z))) (+ z (* x y)))
           (if (<= y 1.95e+144) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * sin(y);
	double t_1 = z * cos(y);
	double tmp;
	if (y <= -6.1e+147) {
		tmp = t_0;
	} else if (y <= -1.15e+103) {
		tmp = t_1;
	} else if (y <= -4000000.0) {
		tmp = t_0;
	} else if (y <= 0.035) {
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	} else if (y <= 1.95e+144) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * sin(y)
    t_1 = z * cos(y)
    if (y <= (-6.1d+147)) then
        tmp = t_0
    else if (y <= (-1.15d+103)) then
        tmp = t_1
    else if (y <= (-4000000.0d0)) then
        tmp = t_0
    else if (y <= 0.035d0) then
        tmp = ((-0.5d0) * (y * (y * z))) + (z + (x * y))
    else if (y <= 1.95d+144) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.sin(y);
	double t_1 = z * Math.cos(y);
	double tmp;
	if (y <= -6.1e+147) {
		tmp = t_0;
	} else if (y <= -1.15e+103) {
		tmp = t_1;
	} else if (y <= -4000000.0) {
		tmp = t_0;
	} else if (y <= 0.035) {
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	} else if (y <= 1.95e+144) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.sin(y)
	t_1 = z * math.cos(y)
	tmp = 0
	if y <= -6.1e+147:
		tmp = t_0
	elif y <= -1.15e+103:
		tmp = t_1
	elif y <= -4000000.0:
		tmp = t_0
	elif y <= 0.035:
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y))
	elif y <= 1.95e+144:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * sin(y))
	t_1 = Float64(z * cos(y))
	tmp = 0.0
	if (y <= -6.1e+147)
		tmp = t_0;
	elseif (y <= -1.15e+103)
		tmp = t_1;
	elseif (y <= -4000000.0)
		tmp = t_0;
	elseif (y <= 0.035)
		tmp = Float64(Float64(-0.5 * Float64(y * Float64(y * z))) + Float64(z + Float64(x * y)));
	elseif (y <= 1.95e+144)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * sin(y);
	t_1 = z * cos(y);
	tmp = 0.0;
	if (y <= -6.1e+147)
		tmp = t_0;
	elseif (y <= -1.15e+103)
		tmp = t_1;
	elseif (y <= -4000000.0)
		tmp = t_0;
	elseif (y <= 0.035)
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	elseif (y <= 1.95e+144)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.1e+147], t$95$0, If[LessEqual[y, -1.15e+103], t$95$1, If[LessEqual[y, -4000000.0], t$95$0, If[LessEqual[y, 0.035], N[(N[(-0.5 * N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.95e+144], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.10000000000000033e147 or -1.15000000000000004e103 < y < -4e6 or 0.035000000000000003 < y < 1.95000000000000009e144

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 50.1%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{\sin y} \cdot \sqrt{\sin y}\right)} + z \cdot \cos y \]

      2. associate-*r* 50.2%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\sin y}\right) \cdot \sqrt{\sin y}} + z \cdot \cos y \]

      3. fma-def 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

   3. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 49.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)}} \]

      2. pow3 49.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)}\right)}^{3}} \]

      3. fma-udef 49.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x \cdot \sqrt{\sin y}\right) \cdot \sqrt{\sin y} + z \cdot \cos y}}\right)}^{3} \]

      4. associate-*r* 49.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \left(\sqrt{\sin y} \cdot \sqrt{\sin y}\right)} + z \cdot \cos y}\right)}^{3} \]

      5. add-sqr-sqrt 97.9%

         \[\leadsto {\left(\sqrt[3]{x \cdot \color{blue}{\sin y} + z \cdot \cos y}\right)}^{3} \]

      6. *-commutative 97.9%

         \[\leadsto {\left(\sqrt[3]{x \cdot \sin y + \color{blue}{\cos y \cdot z}}\right)}^{3} \]

   5. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \sin y + \cos y \cdot z}\right)}^{3}} \]

   6. Taylor expanded in z around 0 66.1%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\sin y \cdot x\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 66.1%

         \[\leadsto \color{blue}{1} \cdot \left(\sin y \cdot x\right) \]

      2. *-lft-identity 66.1%

         \[\leadsto \color{blue}{\sin y \cdot x} \]

   8. Simplified 66.1%

      \[\leadsto \color{blue}{\sin y \cdot x} \]

   if -6.10000000000000033e147 < y < -1.15000000000000004e103 or 1.95000000000000009e144 < y

   1. Initial program 99.5%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 41.2%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{\sin y} \cdot \sqrt{\sin y}\right)} + z \cdot \cos y \]

      2. associate-*r* 41.2%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\sin y}\right) \cdot \sqrt{\sin y}} + z \cdot \cos y \]

      3. fma-def 41.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

   3. Applied egg-rr 41.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 70.5%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -4e6 < y < 0.035000000000000003

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 97.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(y \cdot x + z\right)} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 94.5%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-udef 93.9%

         \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot z\right)} - 1\right)} + \left(y \cdot x + z\right) \]

      3. unpow2 93.9%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right)} - 1\right) + \left(y \cdot x + z\right) \]

      4. associate-*l* 93.9%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(y \cdot z\right)}\right)} - 1\right) + \left(y \cdot x + z\right) \]

   4. Applied egg-rr 93.9%

      \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)} - 1\right)} + \left(y \cdot x + z\right) \]
   5. Step-by-step derivation

      1. expm1-def 94.5%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-log1p 97.8%

         \[\leadsto -0.5 \cdot \color{blue}{\left(y \cdot \left(y \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

   6. Simplified 97.8%

      \[\leadsto -0.5 \cdot \color{blue}{\left(y \cdot \left(y \cdot z\right)\right)} + \left(y \cdot x + z\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+147}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+103}:\\ \;\;\;\;z \cdot \cos y\\ \mathbf{elif}\;y \leq -4000000:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;-0.5 \cdot \left(y \cdot \left(y \cdot z\right)\right) + \left(z + x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 4: 85.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-159} \lor \neg \left(x \leq 1.25 \cdot 10^{-122}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.4e-159) (not (<= x 1.25e-122)))
   (+ z (* x (sin y)))
   (* z (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.4e-159) || !(x <= 1.25e-122)) {
		tmp = z + (x * sin(y));
	} else {
		tmp = z * cos(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-3.4d-159)) .or. (.not. (x <= 1.25d-122))) then
        tmp = z + (x * sin(y))
    else
        tmp = z * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.4e-159) || !(x <= 1.25e-122)) {
		tmp = z + (x * Math.sin(y));
	} else {
		tmp = z * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.4e-159) or not (x <= 1.25e-122):
		tmp = z + (x * math.sin(y))
	else:
		tmp = z * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.4e-159) || !(x <= 1.25e-122))
		tmp = Float64(z + Float64(x * sin(y)));
	else
		tmp = Float64(z * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.4e-159) || ~((x <= 1.25e-122)))
		tmp = z + (x * sin(y));
	else
		tmp = z * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.4e-159], N[Not[LessEqual[x, 1.25e-122]], $MachinePrecision]], N[(z + N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999984e-159 or 1.25e-122 < x

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 86.0%

      \[\leadsto x \cdot \sin y + \color{blue}{z} \]

   if -3.39999999999999984e-159 < x < 1.25e-122

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 54.7%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{\sin y} \cdot \sqrt{\sin y}\right)} + z \cdot \cos y \]

      2. associate-*r* 54.8%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\sin y}\right) \cdot \sqrt{\sin y}} + z \cdot \cos y \]

      3. fma-def 54.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

   3. Applied egg-rr 54.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 95.2%

      \[\leadsto \color{blue}{\cos y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-159} \lor \neg \left(x \leq 1.25 \cdot 10^{-122}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 5: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0275 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(y \cdot \left(y \cdot z\right)\right) + \left(z + x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0275) (not (<= y 0.85)))
   (* z (cos y))
   (+ (* -0.5 (* y (* y z))) (+ z (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0275) || !(y <= 0.85)) {
		tmp = z * cos(y);
	} else {
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-0.0275d0)) .or. (.not. (y <= 0.85d0))) then
        tmp = z * cos(y)
    else
        tmp = ((-0.5d0) * (y * (y * z))) + (z + (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0275) || !(y <= 0.85)) {
		tmp = z * Math.cos(y);
	} else {
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0275) or not (y <= 0.85):
		tmp = z * math.cos(y)
	else:
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0275) || !(y <= 0.85))
		tmp = Float64(z * cos(y));
	else
		tmp = Float64(Float64(-0.5 * Float64(y * Float64(y * z))) + Float64(z + Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0275) || ~((y <= 0.85)))
		tmp = z * cos(y);
	else
		tmp = (-0.5 * (y * (y * z))) + (z + (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0275], N[Not[LessEqual[y, 0.85]], $MachinePrecision]], N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0275000000000000001 or 0.849999999999999978 < y

   1. Initial program 99.7%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 46.7%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{\sin y} \cdot \sqrt{\sin y}\right)} + z \cdot \cos y \]

      2. associate-*r* 46.7%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\sin y}\right) \cdot \sqrt{\sin y}} + z \cdot \cos y \]

      3. fma-def 46.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

   3. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 47.3%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   if -0.0275000000000000001 < y < 0.849999999999999978

   1. Initial program 100.0%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 99.3%

      \[\leadsto \color{blue}{-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(y \cdot x + z\right)} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 95.8%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-udef 95.3%

         \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot z\right)} - 1\right)} + \left(y \cdot x + z\right) \]

      3. unpow2 95.3%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(y \cdot y\right)} \cdot z\right)} - 1\right) + \left(y \cdot x + z\right) \]

      4. associate-*l* 95.3%

         \[\leadsto -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \left(y \cdot z\right)}\right)} - 1\right) + \left(y \cdot x + z\right) \]

   4. Applied egg-rr 95.3%

      \[\leadsto -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)} - 1\right)} + \left(y \cdot x + z\right) \]
   5. Step-by-step derivation

      1. expm1-def 95.8%

         \[\leadsto -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(y \cdot z\right)\right)\right)} + \left(y \cdot x + z\right) \]

      2. expm1-log1p 99.3%

         \[\leadsto -0.5 \cdot \color{blue}{\left(y \cdot \left(y \cdot z\right)\right)} + \left(y \cdot x + z\right) \]

   6. Simplified 99.3%

      \[\leadsto -0.5 \cdot \color{blue}{\left(y \cdot \left(y \cdot z\right)\right)} + \left(y \cdot x + z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0275 \lor \neg \left(y \leq 0.85\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(y \cdot \left(y \cdot z\right)\right) + \left(z + x \cdot y\right)\\ \end{array} \]

Alternative 6: 41.4% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+201}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+79}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.2e+201) (* x y) (if (<= x 2.45e+79) z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e+201) {
		tmp = x * y;
	} else if (x <= 2.45e+79) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.2d+201)) then
        tmp = x * y
    else if (x <= 2.45d+79) then
        tmp = z
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e+201) {
		tmp = x * y;
	} else if (x <= 2.45e+79) {
		tmp = z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.2e+201:
		tmp = x * y
	elif x <= 2.45e+79:
		tmp = z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.2e+201)
		tmp = Float64(x * y);
	elseif (x <= 2.45e+79)
		tmp = z;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.2e+201)
		tmp = x * y;
	elseif (x <= 2.45e+79)
		tmp = z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.2e+201], N[(x * y), $MachinePrecision], If[LessEqual[x, 2.45e+79], z, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1999999999999999e201 or 2.4499999999999999e79 < x

   1. Initial program 99.9%

      \[x \cdot \sin y + z \cdot \cos y \]

   2. Taylor expanded in y around 0 56.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(y \cdot x + z\right)} \]

   3. Taylor expanded in z around 0 42.0%

      \[\leadsto \color{blue}{y \cdot x} \]

   if -3.1999999999999999e201 < x < 2.4499999999999999e79

   1. Initial program 99.8%

      \[x \cdot \sin y + z \cdot \cos y \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 50.6%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{\sin y} \cdot \sqrt{\sin y}\right)} + z \cdot \cos y \]

      2. associate-*r* 50.6%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\sin y}\right) \cdot \sqrt{\sin y}} + z \cdot \cos y \]

      3. fma-def 50.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

   3. Applied egg-rr 50.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

   4. Taylor expanded in x around 0 69.6%

      \[\leadsto \color{blue}{\cos y \cdot z} \]

   5. Taylor expanded in y around 0 42.3%

      \[\leadsto \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+201}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+79}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 52.7% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ z + x \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ z (* x y)))

C

double code(double x, double y, double z) {
	return z + (x * y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z + (x * y)
end function

Java

public static double code(double x, double y, double z) {
	return z + (x * y);
}

Python

def code(x, y, z):
	return z + (x * y)

Julia

function code(x, y, z)
	return Float64(z + Float64(x * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = z + (x * y);
end

Wolfram

code[x_, y_, z_] := N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]

2. Taylor expanded in y around 0 49.7%

   \[\leadsto \color{blue}{y \cdot x + z} \]

3. Final simplification 49.7%

   \[\leadsto z + x \cdot y \]

Alternative 8: 39.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 z)

C

double code(double x, double y, double z) {
	return z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z
end function

Java

public static double code(double x, double y, double z) {
	return z;
}

Python

def code(x, y, z):
	return z

Julia

function code(x, y, z)
	return z
end

MATLAB

function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.8%

   \[x \cdot \sin y + z \cdot \cos y \]
2. Step-by-step derivation

   1. add-sqr-sqrt 49.4%

      \[\leadsto x \cdot \color{blue}{\left(\sqrt{\sin y} \cdot \sqrt{\sin y}\right)} + z \cdot \cos y \]

   2. associate-*r* 49.4%

      \[\leadsto \color{blue}{\left(x \cdot \sqrt{\sin y}\right) \cdot \sqrt{\sin y}} + z \cdot \cos y \]

   3. fma-def 49.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

3. Applied egg-rr 49.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \sqrt{\sin y}, \sqrt{\sin y}, z \cdot \cos y\right)} \]

4. Taylor expanded in x around 0 56.2%

   \[\leadsto \color{blue}{\cos y \cdot z} \]

5. Taylor expanded in y around 0 35.1%

   \[\leadsto \color{blue}{z} \]

6. Final simplification 35.1%

   \[\leadsto z \]

Reproduce

herbie shell --seed 2023258 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))